## Abstract

Quantum correlation and its measurement are essential in exploring fundamental quantum physics problems and developing quantum-enhanced technologies. A quantum correlation may be generated and manipulated in different spaces, which demands different measurement approaches corresponding to the position, time, frequency, and polarization of quantum particles. In addition, after early proof-of-principle demonstrations, it is of great demand to measure quantum correlation in a Hilbert space large enough for real quantum applications. When the number of modes goes up to several hundreds, the single-mode addressing becomes economically unfeasible, and the processing of correlation events with hardware also becomes extremely challenging. Here, we present a general and large-scale measurement approach of the Correlation on the Spatially Mapped Photon-Level Image. The quantum correlations in other spaces are mapped into the position space and are captured by a single-photon-sensitive imaging system. Synthetic methods are developed to suppress noises so that single-photon registrations can be faithfully identified in images. We eventually succeeded in retrieving all the correlations with a big-data technique from tens of millions of images.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Quantum correlation, as one of the unique features of quantum theory, plays a crucial role in quantum information applications. After experiencing quantum evolution, for instance, quantum interference, quantum particles can be correlated in more diverse ways than the classical counterpart. For example, Hong–Ou–Mandel (HOM) interference can reveal the nonclassical bunching properties of photons [1]. These correlation characteristics are crucial in quantum computing [2,3–5], quantum simulation [6–9], and quantum communication [10–13].

In theory, especially in the field of quantum computing, if we managed to send enough entangled photons into plenty of modes and operate their superposition states simultaneously, we would be able to obtain a sufficiently large quantum state space that may enable a higher computational power than classical computers. In practice, it may still be acceptable to place a single-photon detector behind each spatial mode for comparably small systems [9,14–16]. However, it will become both technically challenging and economically unfeasible to address thousands of modes simultaneously with single-photon detectors, thus creating a decisive bottleneck that would prevent from detecting state spaces large enough for real quantum applications.

Thankfully, recent advances in charge-coupled device (CCD) cameras make it possible to directly image spatial output results at a single-photon level [17–24]. HOM interference was also successfully verified by the low-noise correlation detection on two modes with an intensified camera [25].

An alternative and elegant approach presented in this work is to convert the modes from different degrees of freedom into the modes in position, and then measure all the spatial modes accordingly by the large number of units in single-photon-sensitive cameras. We call this general and large-scale measurement approach the Correlation on the Spatially Mapped Photon-Level Image (COSPLI). We have also developed the methods to retrieve the low-noise signal of single-photon registrations and their correlations from tens of millions of images. As an example for applications, we experimentally demonstrate our COSPLI by measuring the spectral correlations [26,27–29] of parametric down-conversion photons.

## 2. EXPERIMENT

COSPLI can generally be achieved in five steps [Fig. 1(a)]. The first step is to identify the target space where the correlation exists. The quantum states of a single photon can be expressed in various eigenvectors with corresponding eigenvalues, such as the position, time, frequency, and polarization. The target space can be whichever space needed to measure. In Fig. 1(a), we use symbols $\{{u}_{1}^{i}\}$ and $\{{u}_{2}^{j}\}$ to represent the target variables. In our experiments, we detect a joint spectrum of correlated photons to demonstrate this technique. Under this circumstance, the target is the spectrum or the frequency space.

It is usually difficult to detect the target variable directly in a large number of modes. Therefore, a mapping operation is necessary to convert the target to position spaces, $\{{x}_{1}^{i}\}$ and $\{{x}_{2}^{j}\}$, which can be measured by a CCD camera. To demonstrate this, we map the frequency information of photons to their positions. In optical system, prisms and gratings are commonly used as spectroscopic devices. To optimize the efficiency and the splitting angle, we chose a blazed grating to transform different frequency components of photons into their corresponding positions. Within a particular wavelength range, the first-order diffracted spot of blazed grating is the brightest, rather than the zero-order reflected spot, which means that the grating loss can be controlled to a relatively low level.

We can divide the target variables into two classes: continuous variables and discrete variables. For continuous variables like the frequency or time, mapping such variables to positions is achievable although it seems difficult. Take time as an example, if the time precision is low, like ${10}^{-3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{s}$ or longer, mechanical-angle-scanning mirrors can help to reflect photons arriving at different times to different places. However, because mechanical devices cannot reach higher precision, the system will be more complicated in situations where higher precision is required. We propose the use of a time-dependent frequency shifter followed by gratings. By continuously sweeping the frequency shift of an electro-optic modulator or acousto-optic modulator, one can map such continuous variables to different spatial coordinates.

As for discrete variables like polarization, a simple polarization beam splitter will map a pair of orthogonal states to different positions. The most recent record of the maximum number of entangled photons is 12 [31], meaning that the ability to measure the correlation among 24 polarization outputs will be quite sufficient. As the physical systems and the operation complexity increase, this number will be dramatically rising in the future. For a schematic graph, see Figs. 1(b)–1(d).

The mapped positions of photons can be either discrete or continuous. For instance, the outputs of a two-dimensional quantum walk from a photonic chip present a discrete spot array. Thus, we are able to directly divide them into several discrete regions according to the location of spots and construct a correlation matrix. As is shown in Fig. 1(a), symbols $\{{x}_{1}^{i}\}$ and $\{{x}_{2}^{j}\}$ represent different components of converted position variables. In our demonstration experiment, the frequency spectra of photons are continuous, and the converted positions therefore are also continuous. However, in the data analysis process, we have to divide the continuous pattern into equal-size segments. As a result, the continuity of data is destroyed to some extent. Hence, we need to restore its inherent continuity at a later stage.

The forth step is to measure the spatially mapped positions with a proper imaging system. Nowadays, intensified charge-coupled devices (ICCD) and intensified scientific complementary metal oxide semiconductors [32] are being well developed to observe single photons. Their high time resolution can be employed as a temporal filter to suppress noises. We retrieve the spatial information from each frame and retrieve the temporal information from the order of frames. We then can perform the last step, which is to calculate the correlation matrix to analyze the spatial and/or temporal correlation.

Figure 1(g) shows several shiny points that can be distinguished from the background. Although they may be the result of dark counts, these events are all considered to be by actual photons. We assume that the noises are generated with a same probability in every pixel of the camera, resulting in a noise matrix $N({\omega}_{s},{\omega}_{i})={p}_{ns}({\omega}_{s}){p}_{ni}({\omega}_{i})$, which means they should just contribute to a contour base of the final correlation matrix as $S({\omega}_{s},{\omega}_{i})+N({\omega}_{s},{\omega}_{i})$. With an optimized criterion, we use a computer program to automatically identify the existence of registered photons and record their positions ${x}_{s}$ and ${x}_{i}$, and then input this correlation information into a correlation matrix $S$ (${\omega}_{s}$, ${\omega}_{i}$) (see Supplement 1).

As an example, we demonstrate a mapping from the frequency correlation to the spatial correlation of photons [see Fig. 1(e)]. Type-II spontaneous parametric down-conversion [33] is applied to generate two frequency-correlated photons, the joint spectrum of which can be written as

## 3. RESULTS

COSPLI is a technique that allows the acquisition of spatial-temporal correlation of photons. In our experiments, both time-independent and time-dependent spatial correlations are demonstrated. The time width of the camera gate determines whether photons in one frame share the correlation or not. In the time-independent scenario, we set the ICCD gate width as 10 μs, which is far longer than the pump laser pulse interval (12.9 ns). Hundreds of photon pairs may appear in one frame; however, on account of many factors of loss, only one or two photons finally register in one frame. In this case, photons detected in the same frame are seldom originally generated by the same pump pulse, which means they are not temporally correlated and therefore have no frequency correlation. The statistic characteristic of these photons will be the same as those of an ensemble, where particles are all independent. The correlation matrix turns out to be simply the product of probability distribution:

where ${p}_{s}({\omega}_{s})$ and ${p}_{i}({\omega}_{i})$ represent the spectrum distribution of signal and idler photons. ${S}_{\text{indep}}$ is the joint spectrum intensity matrix, representing the probability that the signal and idler photon simultaneously possess frequencies ${\omega}_{s}$ and ${\omega}_{i}$, respectively [34]. Unlike the joint spectrum of correlated photons [Eq. (2)], the time-independent one is separable. Figure 2(b) shows the experimental results obtained from 100,000 frames. The perfect similarity with the prediction [Fig. 2(a)] implies that most of shiny points on the screen are real photons rather than dark counts or environmental noise, which can serve as a test of the ICCD single-photon sensitivity.Now we consider the time-dependent scenario, in which only the two correlated photons generated by the same pump pulse appear in the same frame. We synchronize the generation and detection of correlated photons by triggering ICCD with an electric pulse produced by the pump laser oscillator. Another main difference with the time-independent scenario is the gate width. For the purpose of ensuring that every frame contains at most one pair of the correlated photons, the gate width is set as 12.5 ns, which is shorter than the interval between two laser pulses (12.9 ns). In this case, detected photons should share the expected frequency correlation with each other. We put the frames that contain correlation events together in a video (see Visualization 1).

Experimental results from a time-dependent correlation matrix are obtained from ${10}^{7}$ frames [see Figs. 2(c) and 2(d)]. The rate of effective correlation events is approximately ${10}^{-3}$, in which the intrinsically low excitation rate (20%) is an improvable factor. If applying a heralded entanglement source, the rate of effective correlation events will increase substantially [35,36]. There is a tradeoff between the resolution and the count rate, and we choose 10 pixels as the interval between two components, which provide a resolution of 0.74 nm according to the calibrated spectrum distribution.

To demonstrate COSPLI with a different joint spectrum, we use bandpass filters to release the frequency correlation, which is also an important operation to produce high-purity single photons [37]. The transmission spectrum of 3 nm filters centered at 779.5 nm is shown in Fig. 3(a). The two filters lead to a scissor matrix defined as $G({\omega}_{s},{\omega}_{i})={g}_{s}({\omega}_{s}){g}_{i}({\omega}_{i})$, where ${g}_{s}({\omega}_{s})$ and ${g}_{i}({\omega}_{i})$ are the transmission probabilities of the signal and idler photons, respectively. By multiplying the theoretical correlation matrix ${S}_{\mathrm{dep}}({\omega}_{s},{\omega}_{i})$ by $G({\omega}_{s},{\omega}_{i})$ term by term, we can derive the modified theoretical correlation matrix ${S}_{\mathrm{dep}}^{\prime}({\omega}_{s},{\omega}_{i})$ [see Fig. 3(b)]. The measured joint spectrum is shown in Fig. 3(c). In this measurement, the interval between two frequency components is changed to 5 pixels, which means that the resolution becomes 0.37 nm.

The resolution of the correlation matrix is limited by both the grating and the pixels of the camera, which is $1024\times 1024$. In practice, photon spots often take $2\times 2$ or $3\times 3$ pixels of the screen, and we choose this spot size to balance the acceptable frame rate and the resolution.

We can see that, as is shown in Figs. 2(d) and 3(c), the frequency components of signal and idler photons seem to be discrete rather than continuous. It is because we divide the continuous pattern into many segments, leading to the discontinuity of spectrum for both signal and idler photons. In order to restore it, a Fourier filter method is applied to the derived raw data. Since we have discarded all the useless information in the time space, leaving just the joint spectrum, our revised result is more smooth and clear (Fig. 4).

The Schmidt number (SN) can be a good parameter to quantify our results. For the joint spectrum without adding bandpass filters, photons are highly entangled in the frequency domain. After calculating the eigenmodes and corresponding coefficients, the SN is derived. The theoretical SN is 3.519, and the SN of original data is $4.002\pm 0.136$. After a 2D Fourier filter, the SN of revised data is $3.370\pm 0.110$. After adding 3 nm bandpass filters, the entanglement in the frequency domain is released to some extent. Thus, the theoretical SN decreases to 1.419. But at the same time, these 3 nm bandpass filters also block many photons outside of the filter region. The original experimental data are not enough to derive meaningful decomposed eigenmodes because of large statistical fluctuations. Therefore, the Fourier transformation filter is applied to extract the profile of this joint spectrum. The SN of filtered data turns out to be $1.350\pm 0.200$, which is very well consistent with the theoretical value.

One interesting question to ask is if there is any possibility to further measure the complex phase pattern of the joint spectral amplitude. Although the current device deals with an intensity pattern, there have been works that use a spatial light modulator to encode and decode the phase-shaped wave front [38] and use a reference light to reconstruct the 2D phase structures [22]. Such reconstruction of a phase correlation pattern might be considered a second step of our methods.

## 4. DISCUSSION

In summary, we present an approach of mapping and measuring a large-scale photonic correlation with single-photon imaging. We demonstrate this by measuring the joint spectrum of correlated photons with direct imaging. Unlike previous works [39–41], we do not have to move two detectors $m$ and $n$ times, respectively, to obtain a correlation matrix of $m\times n$. Other than spectrum correlation, correlations on other spaces can also be detected after being mapped to position space. The ability to address large-scale correlations in two dimensions may boost the computational power of analog quantum computing by implementing the Boson sampling [2–4] and quantum walk [24] in very large spaces.

In addition to quantum technologies, COSPLI may find applications in exploring some new region of ultra-weak signals [42], like agriculture [43], food chemistry [44], and biomedicine [45]. For example, it has been reported that the neuronal activity has correlation with an ultra-weak photon emission (UPE). Commercial detectors have been able to observe the UPE from a cultured hippocampal slice. However, such research is limited within the detection of accumulation images [46] or a long-time-periods correlation without broader research scopes [47]. Instead, COSPLI may serve as an entirely new technique to reveal the phenomena where correlated events happen simultaneously.

## Funding

National Key R&D Program of China (2017YFA0303700); National Natural Science Foundation of China (NSFC) (61734005, 11761141014, 11690033); Science and Technology Commission of Shanghai Municipality (STCSM) (15QA1402200, 16JC1400405, 17JC1400403); Shanghai Municipal Education Commission (16SG09, 2017-01-07-00-02-E00049); Zhiyuan Scholar Program (ZIRC2016-01); National Young 1000 Talents Plan.

## Acknowledgment

The authors thank Brian Smith, Lijian Zhang, and Jian-Wei Pan for helpful discussions and suggestions.

See Supplement 1 for supporting content.

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